Analysis on Path Spaces over Riemmannian Manifolds with Boundary

By using Hsu's multiplicative functional for the Neumann heat equation, a natural damped gradient operator is defined for the reflecting Brownian motion on compact manifolds with boundary. This operator is linked to quasi-invariant flows in terms of a integration by parts formula, which leads to the standard log-Sobolev inequality for the associated Dirichlet form on the path space.